Monte Carlo Scheme for Asian and Lookback Option Pricing ¶

ZiJian Feng 690127127@qq.com September, 2021

1. Introduction¶

This is a mini project to price exotic options for the appropriate form of payoff, to consider Asian and lookback options, which are path-dependent options. Firstly, we define state variables as:

Mean: $A=mean (S_{i};i\in I)$

Max: $M=max (S_{i};i\in I)$

Min: $m=min (S_{i};i\in I)$

Where, $I$ is a set of sampling index from asset path by a certain period which can be chosen as 1 day,1 week,1 month, 1 quarter,semi-year,1 year and so on. Mean is a function of set to return the mean of its elements Max is a function of set to return the maximum of its elements. Min is a function of set to return the minimum of its elements.

Then, We set $E$ is delivery price, the payoff functions of different exotic options we considered are listed as follow:

option	payoff function
Asian call option with fixed delivery price:	$\max (A-E, 0)$
Asian put option with fixed delivery price:	$\max (E-A, 0)$
Asian call option with floating delivery price:	$\max (A-S_{T}, 0)$
Asian put option with floating delivery price:	$\max (S_{T}-A, 0)$
Lookback call option with fixed delivery price:	$\max (M-E, 0)$
Lookback put option with fixed delivery price:	$\max (E-m, 0)$
Lookback call option with floating delivery price:	$\max (M-S_{T}, 0)$
Lookback put option with floating delivery price:	$\max (S_{T}-m, 0)$

It is hard to find closed-form solutions for prices. More often than not we must solve option-pricing problems by numerical means. The most useful numerical techniques are Monte Carlo simulations and finite-difference methods.

However, path-dependent options are added demension as a state variable so it is difficult to deduce PDE and implement finite-difference methods. Thus, this project mainly consider Monte Carlo scheme.

Monte Carlo methods are any process that consumes random numbers. These are part of computational algorithms which are based on random sampling to obtain numerical results. Monte Carlo methods are proved to be a very valuable and flexible computational tool in finance and is one of the most widely used methods for optimization and numerical integration problems.

These methods are widely used in high dimensional problems. pricing exotics and complex derivatives where closed form solutions are not directly available. Monte Carlo methods are not just adapted in pricing complex derivatives, It is also extensively used in estimating the portfolio risk such as Value-at-Risk and Expected Shortfall and used in the calculation of worst-case scenarios in stress testing. The downside to that is, it is very computational intensive and demanding.

The mathematics that we need to perform a Monte Carlo simulation can be very basic. Use the expected value of the discounted payoff under the risk-neutral density $\mathbb{Q}$

$$V(S, t)=e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}\left[\bf { Payoff } \left(S_{T}\right)\right]$$

Martingale theory can deduce arbitrage-free principle.It is also seen that Martingale theory is consistent with Black-scholes PDE by using Feynman-Kac formula.Thus,justification for the Martingale theory to price is obvious.

The reasons we choose the Monte Carlo scheme are still including:

• Complex path dependency can often be easily incorporated: We do not have to deduce PDEs which can not be understood easily;
• Many contracts can be priced at the same time: We can use a collect of simulated paths to price Asian and lookback options;
• It is computationally quite efficient in high dimensions: We just need to caculate a added state variable which do not cost many time.

2. Numerical Procedure¶

The numerical procedure of Monte Carlo algorithm for option pricing is generally summarized as follow:

• Step1 : Simulate the risk-neutral Brownian motion starting at today’s value of the asset $S_0$ over the required time horizon. This gives one realization of the underlying price path.

• Step2 : For this realization calculate the option payoff. Note that we have to caculate state variables based on simulated price path which are putted into payoff function

• Step3 : Perform many more such realizations over the time horizon.

• Step4 : Calculate the average payoff over all realizations.
• Step5 : Take the present value of this average, this is the option value.

The key procedure is how to simulate the underlying price path. Price process follows the risk-neutral stochastic differential equation which may not have closed-form solutions. A general method that always works is to use a discrete version of the stochastic differential equation for price of which a typical representative is Euler-Maruyama scheme

$$\delta S=r S \delta t+\sigma S \sqrt{\delta t} \phi,$$

where $\phi$ is from a standardized Normal distribution, $r$ is risk-free interest rate

This method has an error of $O(\delta t)$. Because we are only simulating a finite number of an infinite number of possible paths, the error due to using $N$ realizations of the asset price paths is $O(\frac{1}{\sqrt{N}})$.

We set an initial example using the following set of sample data:

• Today's stock price S0 = 100
• Strike E = 100
• Time to expiry (T-t)= 1 year
• volatility = 20%
• constant risk-free interest rate r = 5%

It is necessary to set some numerical parameters:

• $\delta t = \frac{1}{252}$, meaning every day
• $N=100000$, we have balanced between precision and efficiency
• sampling period usually equals 21 if there is no specification

This project is aimed to solve initial example by choose different sampling periods, such as 1 day,1 week,1 month,1 quarter,semi-year,1 year. Then we set sampling period to be 1 month, vary the data to see the affect on the option prices. In the appendix we also make a detailed exploration to analyse error.

3. Coding Preparation¶

It is necessary to set basic configuration before explorate. Some required libraries will be imported.Some functions will be also defined, though may be redefined below by different problem.

Import Required Libraries¶

User Defined Function¶

The function simulate_path is used to simulate the underlying asset price paths. it has 6 aguments:

• s0: initial spot level [float]
• mu: mu = rf in risk neutral framework / year [float]
• sigma: volatility / year [float]
• horizon: time horizion / year [float]
• timesteps: number of time steps [int]
• n_sims: number of simulation [int]

The return is a array shape of which is $(timesteps+1) \times n\_sims$.Its rows represent time, columns represent every simulated path.

The function caculate_exotics_price is used to caculate exotics prices. it has 6 aguments:

• S_path: simulated price path data of which rows represent time, columns represent every simulated path [dataframe]
• sampling_period: $sampling\_period \times \delta t$ is time between two times sampling [int]
• E: delivery price [float]
• r: risk free interest/year [float]
• horizon: time horizion/year [float]

The return is a list containing different extotic option prices.

3. Initial Problem¶

Generate Simulated Path¶

We set inputs according to initial example above. Simulate underlying asset prices and visualize them.

Analyse: From the histogram above, it is clearly seen that price at maturity is subject to right-skewed log-normal distribution. Some extreme data bigger than mean about 100 is dangerous for call option short compared standard normal distribution.

Caculate Exotic Option based on Simulated Price Path¶

Caculate diffrent option prices by sampling by 1 Day, 1 Week, 1 Month, 1 quarter, semi-year,1 year based on the data simulated above.

Analyse: From the table, it clearly seen that:

Firstly, European call and put option is not related to sampling period since their payoff functions are only about final price.

Then, Asian call and put with fixed delivery increase with the increase of sampling period,since the mean function is increasingly volatile when sampling period increases. If the sampling period is 252, the mean is reduced to final price $S_{T}$, so the price of Asian call and put with fixed delivery is same as European call and put option respectively.

However, lookback call and put with fixed delivery has the opposite conclusion: the option prices decrease with the increase of sampling period as a result of stability of maximum and minimum function with sampling period increasing.

Last, All option with float delivery decrease with the increase of sampling period because the mean, maximum and minimum are all reduced to final price, which are equals float delivery $S_{T}$ if the sampling period is 252. Under this situation, their prices equal 0 though there hardly exists option of which sampling period equals time to expiry in practice. We assume this only for mathematic analysis.

4. Vary the Data¶

Change $S_0$¶

Set sampling period to be 21. Caculate diffrent option prices with $S_{0}$ varies from 60 to 150, step is 5.

Analyse: From the figure, it is clearly seen that all call option prices increase with the rise of $S_0$ while all put option prices decrease. Moreover, the change magnitude of options with fixed delivery price is larger than options with floating delivery price.

Change E¶

For European call option, the price is a convex function of the delivery price. We can prove it easily.The payoff function $max\left(S_T-E,0\right)$ is a convex function of the delivery price. since expection integral has property of convexity preserving, so

$C=e^{-r T} \int_{-\infty}^{\infty} max\left(S_T-E,0\right) p\left(S_{T},T,S_0,0\right) d S_{T}$

is also a convex function of the delivery price, where $p\left(S_{T},T,S_0,0\right)$ is transition density function of underlying price.

It is easy to prove that a line joining any two points on the graph of convex function lies on or above the graph. Let ${C(E)}$ denote the call option price as a function of strike ${E}$. Then this is equivalent to saying, for ${E_{1}<E_{2}}$,

${ \theta C\left(E_{1}\right)+(1-\theta) C\left(E_{2}\right) \geq C\left(\theta E_{1}+(1-\theta) E_{2}\right)}$, for all ${0 \leq \theta \leq 1.\quad(*)}$

The important thing to note here is that the final payoff is convex in strike for a fixed value in the underlying, that is the function ${ \max\left(S_{T}-E,0\right) }$ is convex in ${E}$. With this in mind, consider a portfolio long ${\theta}$ call options struck at $E_{1}$, long ${\left(1-\theta\right)}$ options struck at $E_{2}$ and short one option struck at $\theta E_{1}+(1-\theta) E_{2}$ (all options with the same maturity). Since the final payoff is convex in strike, $(*)$ holds at the expiry of the options and we therefore have that our portfolio has non-negative value at expiry. If $C\left(E\right)$ violates convexity at some time before expiry, We can buy/short the portfolio above to make butterfly arbitrage.

The similary conclusion can also be drawed about put option. But what about extotic options? Are their convexities different? We change the $E$ from $50$ to $150$ to see the affect on the option price with setting sampling period is 21.We eliminates options with floating delivery not related to $E$.

Analyse: According the figure above, European options and extotic options are all convex functions of the delivery price from our intuition. Lookback call with fixed delivery price $>$ European call option $>$ Asian call option with fixed delivery price consitently regardless of what E equals. However, for put option, Asian put option with fixed delivery price and European put option violate this consistence.

This give a approxiamte convexity estimate formula:

$convexity=\frac{C^{+}+C^{-}-2C}{ \Delta E^{2}}$

where $C^{+}=C\left(E+\Delta E\right)$,$C^{-}=C\left(E-\Delta E\right)$.

We set $\Delta E=5$, which is not very tiny but still can measure convexity roughly.

Analyse: The results from table above are percentage. For European option and Asian option with fixed delivery price, the convexity of call and put are same. Analogy with the gamma of call and put are same, it can be understood easily. What figure above shows consistents with our experience is that the convexity of European option is greatest when it is at the money. And exotic options are no exceptiones. The data shows that the convexity of Asian option at the money is greater than any others, which implies there may be more butterfly arbitrage profits if opportunity appears.

Change $\sigma$, still Constant Volatility Asumption¶

Set sampling period to be 21. Caculate diffrent option prices with $\sigma$ varies from 0.05 to 0.35, step is 0.01.

Analyse: As we know, option prices rise as volatility rises. From the figure above, vega of lookback options is larger than Asian options because the trend of lookback options is more steep.

Change $\sigma$ , under Stochastic Volatility Asumption¶

Stochastic volatility models take both the underlying price process and volatility process to be stochastic. There are many popular forms for the volatility process, the SABR model, for example, under the future measure, uses

$$\begin{array}{l} d F=\alpha F^{\beta} d W_{1} \\ d \alpha=v \alpha d W_{2} \\ d W_{1} d W_{2}=\rho d t \\ F\left(0\right)=F_0 \\ \alpha\left(0\right)=\alpha_0 \end{array}$$

where $F$ is future price $e^{r(T-t)} S$, $\beta$ is elasticity of Variance, $\alpha$ is stochastic volatility rate, $v$ is the volatility of volatility. $W_{1}$ and $W_{2}$ are Brownian motions with correlation ${\rho}$. $F_0$ and $\alpha_0$ are initial future price and volatility respectively.

For

$$d F=\alpha F^{\beta} d W_{1}\\ $$

note

$$S=e^{-r(T-t)} F\\ $$

by $It\widehat{o}$ lemma,we can get

$$\begin{align} d S=&d e^{-r(T-t)} F\\ =&e^{-r(T-t)} d F+F d e^{-r(T-t)}\\ =&\alpha e^{-r(T-t)} F^{\beta} d W_{1}+r F e^{-r(T-t)} dt\\ =&\alpha e^{(\beta-1) r(T - t)} S^{\beta} d W_{1}+r S dt\\ =&r S d t+\alpha e^{r(\beta-1)(T-t)} S^{\beta} d W_{1} \end{align}$$

When interest rate is not random, $\mathbb{T}=\mathbb{Q}$. We can use the expected value of the discounted payoff under the future measure $\mathbb{T}$(or say $\mathbb{Q}$ )

$$\begin{align} V(S, t)=&e^{-r(T-t)} \mathbb{E}^{\mathbb{T}}\left[\bf { Payoff } \left(F_{T}\right)\right]\\ =&e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}\left[\bf { Payoff } \left(S_{T}\right)\right] \end{align}$$

Redefine Function¶

The function SV_simulate_path is used to simulate the underlying asset price paths under stochastic volatility asumption. it has 8 aguments:

• s0: initial spot level [float]
• mu: mu = rf in risk neutral framework / year [float]
• beta: elasticity of variance coefficient [float]
• v: the volatility of volatility [float]
• alpha: $\alpha_0$ are initial volatility [float]
• horizon: time horizion / year [float]
• timesteps: number of time steps [int]
• n_sims: number of simulation [int]

The return is a array shape of which is $(timesteps+1) \times n\_sims$. Its rows represent time, columns represent every simulated path.

Generate Simulated Path under Stochastic Volatility Asumption¶

Simulate underlying asset prices under stochastic volatility asumption and visualize them.

We set beta=1, v=0.4, alpha=0.2, rho=0.2. Other parameters are the same as default values above.

Next we only let one parameter varies, and others keep the above settings.